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Theorem cantnfres 9413
Description: The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfrescl.d (𝜑𝐷 ∈ On)
cantnfrescl.b (𝜑𝐵𝐷)
cantnfrescl.x ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
cantnfrescl.a (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t 𝑇 = dom (𝐴 CNF 𝐷)
cantnfres.m (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
Assertion
Ref Expression
cantnfres (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfres
Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfrescl.d . . . . . . . . . . . . 13 (𝜑𝐷 ∈ On)
2 cantnfrescl.b . . . . . . . . . . . . 13 (𝜑𝐵𝐷)
3 cantnfrescl.x . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
41, 2, 3extmptsuppeq 7995 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
5 oieq2 9250 . . . . . . . . . . . 12 (((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅) → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
64, 5syl 17 . . . . . . . . . . 11 (𝜑 → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
76fveq1d 6773 . . . . . . . . . 10 (𝜑 → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
873ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
98oveq2d 7287 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = (𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
10 suppssdm 7984 . . . . . . . . . . . . 13 ((𝑛𝐵𝑋) supp ∅) ⊆ dom (𝑛𝐵𝑋)
11 eqid 2740 . . . . . . . . . . . . . . 15 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
1211dmmptss 6143 . . . . . . . . . . . . . 14 dom (𝑛𝐵𝑋) ⊆ 𝐵
1312a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝑛𝐵𝑋) ⊆ 𝐵)
1410, 13sstrid 3937 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
15143ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
16 eqid 2740 . . . . . . . . . . . . . 14 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐵𝑋) supp ∅))
1716oif 9267 . . . . . . . . . . . . 13 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)):dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))⟶((𝑛𝐵𝑋) supp ∅)
1817ffvelrni 6957 . . . . . . . . . . . 12 (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
19183ad2ant2 1133 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
2015, 19sseldd 3927 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ 𝐵)
2120fvresd 6791 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
2223ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵𝐷)
2322resmptd 5947 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋) ↾ 𝐵) = (𝑛𝐵𝑋))
2423fveq1d 6773 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
258fveq2d 6775 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
2621, 24, 253eqtr3d 2788 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
279, 26oveq12d 7289 . . . . . . 7 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) = ((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))))
2827oveq1d 7286 . . . . . 6 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧) = (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧))
2928mpoeq3dva 7346 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
306dmeqd 5813 . . . . . 6 (𝜑 → dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
31 eqid 2740 . . . . . 6 On = On
32 mpoeq12 7342 . . . . . 6 ((dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) ∧ On = On) → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
3330, 31, 32sylancl 586 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
3429, 33eqtrd 2780 . . . 4 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)))
35 eqid 2740 . . . 4 ∅ = ∅
36 seqomeq12 8276 . . . 4 (((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
3734, 35, 36sylancl 586 . . 3 (𝜑 → seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅))
3837, 30fveq12d 6778 . 2 (𝜑 → (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
39 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
40 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
41 cantnfs.b . . 3 (𝜑𝐵 ∈ On)
42 cantnfres.m . . 3 (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
43 eqid 2740 . . 3 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
4439, 40, 41, 16, 42, 43cantnfval2 9405 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))))
45 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
46 eqid 2740 . . 3 OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅))
47 cantnfrescl.a . . . . 5 (𝜑 → ∅ ∈ 𝐴)
4839, 40, 41, 1, 2, 3, 47, 45cantnfrescl 9412 . . . 4 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
4942, 48mpbid 231 . . 3 (𝜑 → (𝑛𝐷𝑋) ∈ 𝑇)
50 eqid 2740 . . 3 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)
5145, 40, 1, 46, 49, 50cantnfval2 9405 . 2 (𝜑 → ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)) = (seqω((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴o (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·o ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
5238, 44, 513eqtr4d 2790 1 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1542  wcel 2110  Vcvv 3431  cdif 3889  wss 3892  c0 4262  cmpt 5162   E cep 5495  dom cdm 5590  cres 5592  Oncon0 6265  cfv 6432  (class class class)co 7271  cmpo 7273   supp csupp 7968  seqωcseqom 8269   +o coa 8285   ·o comu 8286  o coe 8287  OrdIsocoi 9246   CNF ccnf 9397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-supp 7969  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-seqom 8270  df-1o 8288  df-oadd 8292  df-er 8481  df-map 8600  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-fsupp 9107  df-oi 9247  df-cnf 9398
This theorem is referenced by: (None)
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